The Stacks project

Lemma 4.18.4. Let $\mathcal{C}$ be a category. The following are equivalent:

  1. Finite limits exist in $\mathcal{C}$.

  2. Finite products and equalizers exist.

  3. The category has a final object and fibred products exist.

Proof. Since products of pairs, fibre products, equalizers, and final objects are limits over finite index categories we see that (1) implies both (2) and (3). By Lemma 4.14.10 above we see that (2) implies (1). Assume (3). Note that the product $A \times B$ is the fibre product over the final object. If $a, b : A \to B$ are morphisms of $\mathcal{C}$, then the equalizer of $a, b$ is

\[ (A \times _{a, B, b} A)\times _{(pr_1, pr_2), A \times A, \Delta } A. \]

Thus (3) implies (2) and the lemma is proved. $\square$


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